A model of syllogistic reasoning as communication.

The reasoner imagines an experimenter who chooses an argument (i.e. premises) conditioned on the reasoner inferring an intended conclusion.

A syllogism is a two-sentence (premise) argument. Each sentence consists of 2 terms, and between the two sentences, 1 of the terms is shared. For example,

B - A

C - B

In this argument, B is the shared term. The task for the reasoner is to generate a conclusion relating the “end-terms”– A & C.

The relations between the terms are quantifiers. In classical syllogisms, the quantifiers are those from Aristotle’s “square of opposition”: {all, some, none, not-all}.

For example,

No B are A

All C are B

{All, Some, No, Not-all} C are A

In this model, the reasoner imagines these sentences apply to concrete situations which are composed of objects with properties. Sentences (premises or conclusion) are then either true or false of a given situation.

Note: The model presented below is for transparency and/or pedagogical purposes. The program itself is computationally expensive. The state space grows exponentially with the number-of-objects parameter. However, we are interested not in the distribution of objects per se, but in the distribution of true-sentences that those objects imply. As such, we can derive an equivalence-class representation of the situation space (objects comprise situations), which is projection of the distribution of situations onto the sentences of interest. The equivalence-class of situations does not grow with the number of objects and is much faster to run. However, the number-of-objects and base-rate parameters cannot be changed inside the model. The version of that model (with best-fit parameters n_objects = 5, br = 0.25 used in Ref:tessler2014syllogisms) can be found here.

``````(define all-true (lambda (lst) (apply and lst)))
(define some-true (lambda (lst) (apply or lst)))

; assume the situations constructed have at least one object with each of the properties
(define existential-import (lambda (A B objects)
(and (some-true (map A objects)) (some-true (map B objects)))))

(define all (lambda (A B)
(if (existential-import A B objects)
(all-true (map (lambda (x) (if (A x) (B x) true))
objects))
false)))

(define some (lambda (A B)
(if (existential-import A B objects)
(some-true (map (lambda (x) (if (A x) (B x) false))
objects))
false)))

(define none (lambda (A B)
(if (existential-import A B objects)
(all-true (map (lambda (x) (if (A x) (not (B x)) true))
objects))
false)))

(define not-all (lambda (A B)
(if (existential-import A B objects)
(some-true (map (lambda (x) (if (A x) (not (B x)) false))
objects))
false)))

(define (raise-to-power dist alph)
(list (first dist) (map (lambda (x) (pow x alph)) (second dist))))

; pass strings, which then call functions of the same name
(define (meaning word)
(case word
(('all) all)
(('some) some)
(('not-all) not-all)
(('none) none)))

; the reasoner has uninformative prior beliefs about what the conclusion should be
(define (conclusion-prior) (uniform-draw (list 'all 'some 'not-all 'none)))
; rhe reasoner has uninformative prior beliefs about what arguments the experimenter could give
(define (premise-prior) (uniform-draw (list 'all 'some 'not-all 'none)))

; parameter 1: the reasoner's prior beliefs about the rarity of properties
(define br 0.25)
; parameter 2: number of objects in the situation the reasoner imagines
(define objects (list 'o1 'o2 'o3))

(define argument-strength
(mem
; argument-strength is a function of the argument's premises
(lambda (premise-one premise-two)
(enumeration-query
; properties map objects to truth-values i.e. whether or not the object has the property
; together with the list of objects, these represent situations over which reasoning occurs
(define A (mem (lambda (x) (flip br))))
(define B (mem (lambda (x) (flip br))))
(define C (mem (lambda (x) (flip br))))
; the reasoner is also uncertain about what conclusion is true
(define conclusion (conclusion-prior))

; what do we want to know? what is the conclusion.
conclusion

; condition on the syllogism applying to the situation
; notice the form of the syllogism present in this part of the query
(and
; the premise quantifiers apply to: A-B & B-C
((meaning premise-one) A B)
((meaning premise-two) B C)
; the conclusion quantifier is true of terms: A & C
((meaning conclusion) A C)
)))))

(define experimenter
(mem
; the experimenter takes in the conclusion as an argument (i.e. he has a conclusion in mind)
(lambda (conclusion)
(enumeration-query
; the experimenter draws premises from the premise-prior (uniform) distribution
(define premise-one (premise-prior))
(define premise-two (premise-prior))

; the experimenter produces two premises
(list premise-one premise-two)

; the experimenter wants the reasoner to draw a particular conclusion, gives the premises
; parameter 3: "optimality" -- the degree to which the experimenter's argument is optimal for the conclusion
(equal? conclusion (apply multinomial (raise-to-power (argument-strength premise-one premise-two) 4.75)))))))

(define pragmatic-reasoner
(mem
; the reasoner takes in two premises as arguments
(lambda (premise-one premise-two)
(enumeration-query
; the pragmatic reasoner has the exact same generative model as the argument strength model
(define A (mem (lambda (x) (flip br))))
(define B (mem (lambda (x) (flip br))))
(define C (mem (lambda (x) (flip br))))
(define conclusion (conclusion-prior))

; the reasoner produces a conclusion
conclusion

(and
; the conclusion quantifier is true of terms: A & C
((meaning conclusion) A C)
; the premises now serve a different role; the premises are imagined to come from an experimenter
; who has a particular conclusion in mind
(equal? (list premise-one premise-two) (apply multinomial (experimenter conclusion))))
))))

; All A are B
; No B are C [cogsci paper, Fig 2 [1]]

;(pragmatic-reasoner 'all 'none)
(argument-strength 'all 'none)

; this is a valid syllogism with 2 valid conclusions
; using the argument-strength alone produces no preference among valid conclusions
; the pragmatic reasonser draws the conclusion not only true of the situation
; but also which was intended to be drawn, inferring that "None" is the more likely intended conclusion
``````

References:

• Cite:tessler2014syllogisms